Optimal. Leaf size=109 \[ -\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{4 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d} \]
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Rubi [A] time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {\csc ^{11}(c+d x)}{11 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}-\frac {\csc ^8(c+d x)}{4 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^{12} (a-x)^3 (a+x)^2}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {a^5}{x^{12}}-\frac {a^4}{x^{11}}-\frac {2 a^3}{x^{10}}+\frac {2 a^2}{x^9}+\frac {a}{x^8}-\frac {1}{x^7}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}-\frac {\csc ^8(c+d x)}{4 a d}+\frac {2 \csc ^9(c+d x)}{9 a d}+\frac {\csc ^{10}(c+d x)}{10 a d}-\frac {\csc ^{11}(c+d x)}{11 a d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 68, normalized size = 0.62 \[ \frac {\csc ^6(c+d x) \left (-1260 \csc ^5(c+d x)+1386 \csc ^4(c+d x)+3080 \csc ^3(c+d x)-3465 \csc ^2(c+d x)-1980 \csc (c+d x)+2310\right )}{13860 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 128, normalized size = 1.17 \[ \frac {1980 \, \cos \left (d x + c\right )^{4} - 880 \, \cos \left (d x + c\right )^{2} - 231 \, {\left (10 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 160}{13860 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 66, normalized size = 0.61 \[ \frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 69, normalized size = 0.63 \[ \frac {\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {2}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{11 \sin \left (d x +c \right )^{11}}+\frac {1}{10 \sin \left (d x +c \right )^{10}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 66, normalized size = 0.61 \[ \frac {2310 \, \sin \left (d x + c\right )^{5} - 1980 \, \sin \left (d x + c\right )^{4} - 3465 \, \sin \left (d x + c\right )^{3} + 3080 \, \sin \left (d x + c\right )^{2} + 1386 \, \sin \left (d x + c\right ) - 1260}{13860 \, a d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.04, size = 65, normalized size = 0.60 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^5}{6}-\frac {{\sin \left (c+d\,x\right )}^4}{7}-\frac {{\sin \left (c+d\,x\right )}^3}{4}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {\sin \left (c+d\,x\right )}{10}-\frac {1}{11}}{a\,d\,{\sin \left (c+d\,x\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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